In this article, we consider general linear systems whose right-hand sides are represented by the Riemann–Stieltjes integral with kernels given by a matrix function of bounded variation. We establish some minimal requirements on the convergence of the kernels that ensure the convergence of Lyapunov matrices of the perturbed systems. Using functional analytic approach, we analyze the continuity of the dependence of Lyapunov matrices on the right-hand sides. Results obtained in this paper improve the previously known results for multiple delay systems and distributed delay systems. One significant advantage of our approach is that no assumption on the exponential stability is being made.